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Photo by Vickie Kelly, 2007. Greg Kelly, Hanford High School, Richland, Washington. 9.5 Testing Convergence at Endpoints. Petrified Forest National Park, Arizona. If is a series with positive terms and. then:. The series converges if .

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Presentation Transcript
slide1

Photo by Vickie Kelly, 2007

Greg Kelly, Hanford High School, Richland, Washington

9.5 Testing Convergence at Endpoints

Petrified Forest National Park,

Arizona

slide2

If is a series with positive terms and

then:

The series converges if .

The series diverges if .

The test is inconclusive if .

Remember:

The Ratio Test:

slide3

This section in the book presents several other tests or techniques to test for convergence, and discusses some specific convergent and divergent series.

slide4

If is a series with positive terms and

then:

The series converges if .

The series diverges if .

The test is inconclusive if .

Nth Root Test:

Note that the rules are the same as for the Ratio Test.

slide6

formula #104

formula #103

Indeterminate, so we use L’Hôpital’s Rule

slide7

it converges

example:

?

slide8

it diverges

another example:

slide9

The Integral Test

If is a positive sequence and where

is a continuous, positive decreasing function, then:

and both converge or both diverge.

Remember that when we first studied integrals, we used a summation of rectangles to approximate the area under a curve:

This leads to:

slide10

Example 1:

Does converge?

Since the integral converges, the series must converge.

(but not necessarily to 2.)

slide11

converges if , diverges if .

p-series Test

We could show this with the integral test.

If this test seems backward after the ratio and nth root tests, remember that larger values of p would make the denominators increase faster and the terms decrease faster.

slide12

It diverges very slowly, but it diverges.

Because the p-series is so easy to evaluate, we use it to compare to other series.

the harmonic series:

diverges.

(It is a p-series with p=1.)

slide13

If , then both and

converge or both diverge.

If , then converges if converges.

If , then diverges if diverges.

Limit Comparison Test

If and for all (N a positive integer)

slide14

Since diverges, the series diverges.

Example 3a:

When n is large, the function behaves like:

harmonic series

slide15

Since converges, the series converges.

Example 3b:

When n is large, the function behaves like:

geometric series

slide16

Alternating Series Test

If the absolute values of the terms approach zero, then an alternating series will always converge!

Alternating Series

The signs of the terms alternate.

Good news!

example:

This series converges (by the Alternating Series Test.)

This series is convergent, but not absolutely convergent.

Therefore we say that it is conditionally convergent.

slide17

Alternating Series Estimation Theorem

For a convergent alternating series, the truncation error is less than the first missing term, and is the same sign as that term.

Since each term of a convergent alternating series moves the partial sum a little closer to the limit:

This is a good tool to remember, because it is easier than the LaGrange Error Bound.

slide18

F3

4

There is a flow chart on page 505 that might be helpful for deciding in what order to do which test. Mostly this just takes practice.

To do summations on the TI-89:

becomes

becomes

slide19

ENTER

ENTER

GRAPH

Y=

WINDOW

To graph the partial sums, we can use sequence mode.

MODE

Graph…….

4

slide20

ENTER

ENTER

GRAPH

Y=

WINDOW

Table

To graph the partial sums, we can use sequence mode.

MODE

Graph…….

4

slide21

ENTER

ENTER

GRAPH

Y=

WINDOW

Table

To graph the partial sums, we can use sequence mode.

MODE

Graph…….

4

p